Optimal. Leaf size=486 \[ \frac{44 b^{21/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (3 b B-5 A c) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{16575 c^{19/4} \sqrt{b x^2+c x^4}}-\frac{8 b^2 x^{9/2} \sqrt{b x^2+c x^4} (3 b B-5 A c)}{7735 c^2}+\frac{88 b^3 x^{5/2} \sqrt{b x^2+c x^4} (3 b B-5 A c)}{69615 c^3}+\frac{88 b^5 x^{3/2} \left (b+c x^2\right ) (3 b B-5 A c)}{16575 c^{9/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{88 b^4 \sqrt{x} \sqrt{b x^2+c x^4} (3 b B-5 A c)}{49725 c^4}-\frac{88 b^{21/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (3 b B-5 A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{16575 c^{19/4} \sqrt{b x^2+c x^4}}-\frac{4 b x^{13/2} \sqrt{b x^2+c x^4} (3 b B-5 A c)}{595 c}-\frac{2 x^{9/2} \left (b x^2+c x^4\right )^{3/2} (3 b B-5 A c)}{105 c}+\frac{2 B x^{5/2} \left (b x^2+c x^4\right )^{5/2}}{25 c} \]
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Rubi [A] time = 0.670511, antiderivative size = 486, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2039, 2021, 2024, 2032, 329, 305, 220, 1196} \[ -\frac{8 b^2 x^{9/2} \sqrt{b x^2+c x^4} (3 b B-5 A c)}{7735 c^2}+\frac{88 b^3 x^{5/2} \sqrt{b x^2+c x^4} (3 b B-5 A c)}{69615 c^3}+\frac{88 b^5 x^{3/2} \left (b+c x^2\right ) (3 b B-5 A c)}{16575 c^{9/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{88 b^4 \sqrt{x} \sqrt{b x^2+c x^4} (3 b B-5 A c)}{49725 c^4}+\frac{44 b^{21/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (3 b B-5 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{16575 c^{19/4} \sqrt{b x^2+c x^4}}-\frac{88 b^{21/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (3 b B-5 A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{16575 c^{19/4} \sqrt{b x^2+c x^4}}-\frac{4 b x^{13/2} \sqrt{b x^2+c x^4} (3 b B-5 A c)}{595 c}-\frac{2 x^{9/2} \left (b x^2+c x^4\right )^{3/2} (3 b B-5 A c)}{105 c}+\frac{2 B x^{5/2} \left (b x^2+c x^4\right )^{5/2}}{25 c} \]
Antiderivative was successfully verified.
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Rule 2039
Rule 2021
Rule 2024
Rule 2032
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int x^{7/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx &=\frac{2 B x^{5/2} \left (b x^2+c x^4\right )^{5/2}}{25 c}-\frac{\left (2 \left (\frac{15 b B}{2}-\frac{25 A c}{2}\right )\right ) \int x^{7/2} \left (b x^2+c x^4\right )^{3/2} \, dx}{25 c}\\ &=-\frac{2 (3 b B-5 A c) x^{9/2} \left (b x^2+c x^4\right )^{3/2}}{105 c}+\frac{2 B x^{5/2} \left (b x^2+c x^4\right )^{5/2}}{25 c}-\frac{(2 b (3 b B-5 A c)) \int x^{11/2} \sqrt{b x^2+c x^4} \, dx}{35 c}\\ &=-\frac{4 b (3 b B-5 A c) x^{13/2} \sqrt{b x^2+c x^4}}{595 c}-\frac{2 (3 b B-5 A c) x^{9/2} \left (b x^2+c x^4\right )^{3/2}}{105 c}+\frac{2 B x^{5/2} \left (b x^2+c x^4\right )^{5/2}}{25 c}-\frac{\left (4 b^2 (3 b B-5 A c)\right ) \int \frac{x^{15/2}}{\sqrt{b x^2+c x^4}} \, dx}{595 c}\\ &=-\frac{8 b^2 (3 b B-5 A c) x^{9/2} \sqrt{b x^2+c x^4}}{7735 c^2}-\frac{4 b (3 b B-5 A c) x^{13/2} \sqrt{b x^2+c x^4}}{595 c}-\frac{2 (3 b B-5 A c) x^{9/2} \left (b x^2+c x^4\right )^{3/2}}{105 c}+\frac{2 B x^{5/2} \left (b x^2+c x^4\right )^{5/2}}{25 c}+\frac{\left (44 b^3 (3 b B-5 A c)\right ) \int \frac{x^{11/2}}{\sqrt{b x^2+c x^4}} \, dx}{7735 c^2}\\ &=\frac{88 b^3 (3 b B-5 A c) x^{5/2} \sqrt{b x^2+c x^4}}{69615 c^3}-\frac{8 b^2 (3 b B-5 A c) x^{9/2} \sqrt{b x^2+c x^4}}{7735 c^2}-\frac{4 b (3 b B-5 A c) x^{13/2} \sqrt{b x^2+c x^4}}{595 c}-\frac{2 (3 b B-5 A c) x^{9/2} \left (b x^2+c x^4\right )^{3/2}}{105 c}+\frac{2 B x^{5/2} \left (b x^2+c x^4\right )^{5/2}}{25 c}-\frac{\left (44 b^4 (3 b B-5 A c)\right ) \int \frac{x^{7/2}}{\sqrt{b x^2+c x^4}} \, dx}{9945 c^3}\\ &=-\frac{88 b^4 (3 b B-5 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{49725 c^4}+\frac{88 b^3 (3 b B-5 A c) x^{5/2} \sqrt{b x^2+c x^4}}{69615 c^3}-\frac{8 b^2 (3 b B-5 A c) x^{9/2} \sqrt{b x^2+c x^4}}{7735 c^2}-\frac{4 b (3 b B-5 A c) x^{13/2} \sqrt{b x^2+c x^4}}{595 c}-\frac{2 (3 b B-5 A c) x^{9/2} \left (b x^2+c x^4\right )^{3/2}}{105 c}+\frac{2 B x^{5/2} \left (b x^2+c x^4\right )^{5/2}}{25 c}+\frac{\left (44 b^5 (3 b B-5 A c)\right ) \int \frac{x^{3/2}}{\sqrt{b x^2+c x^4}} \, dx}{16575 c^4}\\ &=-\frac{88 b^4 (3 b B-5 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{49725 c^4}+\frac{88 b^3 (3 b B-5 A c) x^{5/2} \sqrt{b x^2+c x^4}}{69615 c^3}-\frac{8 b^2 (3 b B-5 A c) x^{9/2} \sqrt{b x^2+c x^4}}{7735 c^2}-\frac{4 b (3 b B-5 A c) x^{13/2} \sqrt{b x^2+c x^4}}{595 c}-\frac{2 (3 b B-5 A c) x^{9/2} \left (b x^2+c x^4\right )^{3/2}}{105 c}+\frac{2 B x^{5/2} \left (b x^2+c x^4\right )^{5/2}}{25 c}+\frac{\left (44 b^5 (3 b B-5 A c) x \sqrt{b+c x^2}\right ) \int \frac{\sqrt{x}}{\sqrt{b+c x^2}} \, dx}{16575 c^4 \sqrt{b x^2+c x^4}}\\ &=-\frac{88 b^4 (3 b B-5 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{49725 c^4}+\frac{88 b^3 (3 b B-5 A c) x^{5/2} \sqrt{b x^2+c x^4}}{69615 c^3}-\frac{8 b^2 (3 b B-5 A c) x^{9/2} \sqrt{b x^2+c x^4}}{7735 c^2}-\frac{4 b (3 b B-5 A c) x^{13/2} \sqrt{b x^2+c x^4}}{595 c}-\frac{2 (3 b B-5 A c) x^{9/2} \left (b x^2+c x^4\right )^{3/2}}{105 c}+\frac{2 B x^{5/2} \left (b x^2+c x^4\right )^{5/2}}{25 c}+\frac{\left (88 b^5 (3 b B-5 A c) x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{16575 c^4 \sqrt{b x^2+c x^4}}\\ &=-\frac{88 b^4 (3 b B-5 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{49725 c^4}+\frac{88 b^3 (3 b B-5 A c) x^{5/2} \sqrt{b x^2+c x^4}}{69615 c^3}-\frac{8 b^2 (3 b B-5 A c) x^{9/2} \sqrt{b x^2+c x^4}}{7735 c^2}-\frac{4 b (3 b B-5 A c) x^{13/2} \sqrt{b x^2+c x^4}}{595 c}-\frac{2 (3 b B-5 A c) x^{9/2} \left (b x^2+c x^4\right )^{3/2}}{105 c}+\frac{2 B x^{5/2} \left (b x^2+c x^4\right )^{5/2}}{25 c}+\frac{\left (88 b^{11/2} (3 b B-5 A c) x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{16575 c^{9/2} \sqrt{b x^2+c x^4}}-\frac{\left (88 b^{11/2} (3 b B-5 A c) x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{b}}}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{16575 c^{9/2} \sqrt{b x^2+c x^4}}\\ &=\frac{88 b^5 (3 b B-5 A c) x^{3/2} \left (b+c x^2\right )}{16575 c^{9/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{88 b^4 (3 b B-5 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{49725 c^4}+\frac{88 b^3 (3 b B-5 A c) x^{5/2} \sqrt{b x^2+c x^4}}{69615 c^3}-\frac{8 b^2 (3 b B-5 A c) x^{9/2} \sqrt{b x^2+c x^4}}{7735 c^2}-\frac{4 b (3 b B-5 A c) x^{13/2} \sqrt{b x^2+c x^4}}{595 c}-\frac{2 (3 b B-5 A c) x^{9/2} \left (b x^2+c x^4\right )^{3/2}}{105 c}+\frac{2 B x^{5/2} \left (b x^2+c x^4\right )^{5/2}}{25 c}-\frac{88 b^{21/4} (3 b B-5 A c) x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{16575 c^{19/4} \sqrt{b x^2+c x^4}}+\frac{44 b^{21/4} (3 b B-5 A c) x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{16575 c^{19/4} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.214981, size = 160, normalized size = 0.33 \[ \frac{2 \sqrt{x} \sqrt{x^2 \left (b+c x^2\right )} \left (385 b^4 (3 b B-5 A c) \, _2F_1\left (-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{b}\right )-\left (b+c x^2\right )^2 \sqrt{\frac{c x^2}{b}+1} \left (-55 b^2 c \left (35 A+39 B x^2\right )+65 b c^2 x^2 \left (55 A+51 B x^2\right )-221 c^3 x^4 \left (25 A+21 B x^2\right )+1155 b^3 B\right )\right )}{116025 c^4 \sqrt{\frac{c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 518, normalized size = 1.1 \begin{align*} -{\frac{2}{348075\, \left ( c{x}^{2}+b \right ) ^{2}{c}^{5}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( -13923\,B{x}^{14}{c}^{7}-16575\,A{x}^{12}{c}^{7}-31824\,B{x}^{12}b{c}^{6}-39000\,A{x}^{10}b{c}^{6}-18369\,B{x}^{10}{b}^{2}{c}^{5}-23325\,A{x}^{8}{b}^{2}{c}^{5}+72\,B{x}^{8}{b}^{3}{c}^{4}+200\,A{x}^{6}{b}^{3}{c}^{4}-120\,B{x}^{6}{b}^{4}{c}^{3}+4620\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){b}^{6}c-2310\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){b}^{6}c-2772\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){b}^{7}+1386\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){b}^{7}-440\,A{x}^{4}{b}^{4}{c}^{3}+264\,B{x}^{4}{b}^{5}{c}^{2}-1540\,A{x}^{2}{b}^{5}{c}^{2}+924\,B{x}^{2}{b}^{6}c \right ){x}^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}{\left (B x^{2} + A\right )} x^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (B c x^{9} +{\left (B b + A c\right )} x^{7} + A b x^{5}\right )} \sqrt{c x^{4} + b x^{2}} \sqrt{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}{\left (B x^{2} + A\right )} x^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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